Combinatorial Games on Graphs, Coxeter-Dynkin

diagrams, and theGeometry of Root Systems

N J WildbergerSchool of Mathematics and

StatisticsUNSW Sydney

Graphs and Groups,Representations and

Relations

Novosibirsk August 6-19, 2018

All other graphs. . .

Sp(X) < 2

Sp(X) = 2

Sp(X) > 2

ADE graphs ADE ~ graphs

Disclosure: I do not believe in:

a) “irrational numbers”b) “infinite processes that can be completed”c) “infinite sets”d) “axioms” as a basis for mathematics

Since “eigenvalues” are problematic, we need alternate ways to describe theADE / ADE~ / other division.

The SP(X)=2 Perron Frobenius vectors on ADE~ graphs

ADE graphs and Platonic solids

A_n: Cyclic {1,1,n}D_n: Dihedral {2,2,n-2}E_6: Tetrahedral {2,3,3}E_7: Octahedral / Cube {2,3,4}E_8: Icosahedron / Dodecahedron {2,3,5}

{p,q,r} : p faces around an edge, q edges around a vertex, r vertices around a face (or dual)

ADE graphs and simple Lie algebras

W. Killing (1888) classified simple Lie algebras

A_n: sl(n)B_n: so(2n-1)C_n: sp(2n)D_n: so(2n)

E_6: dim = 78E_7: dim = 133E_8: dim = 248F_4: dim = 52G_2: dim = 14

Simple Lie algebras Root systems Dynkin diagrams

Lie groups, symmetric spaces, reflection groups

Simple Lie algebras Lie groups Symmetric spaces

Weyl groups (generalizations of S_n)

Coxeter groups (generated by reflections)

Two other occurrences of ADE graphs

Pierre Gabriel (1972): quivers of finite type and indecomposable representations

John McKay (1979): ADE graphs correspond to finite subgroups of SU(2) / unit quaternions

The 600-cell (E_8)

1+2+3+4+5+6+4+2+3 = 120 2 2 2 2 2 2 2 2 2

Many other occurrences of ADE graphs !!

Von Neumann algebras and II-1 factors

Conformal field theory and Wess-Zumino-Witten models (fusion rule algebras) String theory!

Catastrophe theory

Simple singularites of holomorphic functions (V I Arnold)

Combinatorics!!

The Mutation GameX = simple, connected graph. A population on X is an integer valued function on the vertices.P(X) = populations on X

The mutation s_y at the vertex y: fixes all population values except that at y, which gets replaced by its negative plus the sum of its neighbours.

E6 mutation sequenceFrom a singleton population to the maximal population

Root Populations / roots

A root population is a population obtained from a singleton population by applying any sequence of mutations. R(X) denotes the root populations of the graph X.

R(D4) = R (D4) + R (D4)+ -

Where a root is positive if all its entries are positive (>=0).

ADE GraphsTheorem: R(X) is finite precisely when X is an ADE graph, i.e. in this following list:

These sets R(X) turn out to be the irreducible (simply laced) root systems studied by E. Dynkin.

These are sets of vectors in a Euclidean space satisfying symmetry wrt reflections in hyperplanes

Proof:1: If X is ADE then R(X) is finite (enumerate them!)

2. If X is not ADE, then it contains an ADE~ subgraph

3. Show that if X is ADE~ then R(X) is unbounded: use the Perron Frobenius vector which is unchanged by mutations, for example for E6:

An important conjecture

The Mutation Fact / Conjecture: For any simple connected graph X, R(X) is always the disjoint union of positive and negative roots.

This is known as a consequence of the theory of Coxetergroups, generated by reflections. (Personal communication with Bob Howlett). However we do not have a combinatorial proof/ understanding.

A restatement: a root population can never have both strictly positive and strictly negative entries, for any graph X !!

Positive rootpopulationsof E6

|R(E6)|=36+36=72

A2: The two-dimensional mutation representation ofW(A2) = S_3 = < s1 , s2 >

(a , b)

(- a + b , b)

(- a + b , - a)

(- b , - a)

(a , a - b)

(- b , a - b)

s2

s1 s2 s1 = s2 s1 s2

s1

1 00 1

( )

0 -1-1 0

-1 01 1

-1 -11 0

1 10 -1

( )

( )0 1

-1 -1

( )

( )

( )

e

s2

s2

s1

s1

Braid relation

S1^2 = s2^2 = 1

Reflection relation

A3: The three-dimensional mutation representation ofW(A3) = S_4 = < s1 , s2, s3 > and the Permutahedron

2

12

22

2

121

11

11

11

11

111

1112

13

13

13

133

2

2

1313

13

1313

13

13

11

1

2

211

2

(a , b , c)

(a , a-b+c , c)

(-b+c, a-b+c, c)

(c-b, -a+c, c)

(c-b, -a+c, -a)

(c-b, -b, -a)

11

(-c, -b, -a)

Longest word in W

The Tits Quadratic FormDefine a symmetric bilinear form on P(X) = populations on X via the symmetric matrix C = 2I-A [Cartan matrix] where A is the adjacency of the graph.

𝑄𝑄 𝑎𝑎, 𝑏𝑏, 𝑐𝑐,𝑑𝑑 = 2𝑎𝑎2 + 2𝑏𝑏2 + 2𝑐𝑐2 + 2𝑑𝑑2 − 𝑎𝑎𝑏𝑏 − 𝑏𝑏𝑐𝑐 − 𝑏𝑏𝑑𝑑

Theorem: The mutations s_x are isometries with respect to the Tits quadratic form. So W = < s_x > is a group of isometries.

A2 lattice, sphere Q(v)=2 and root system

Q((a,b))=2a^2+2b^2-ab

Rational TrigonometryA symmetric bilinear form gives geometry! The algebraic approach forgets about distances and angles, and uses quadrance and spread!

Extends to general quadratic forms!

Old Babylonian Trigonometry

Plimpton 322 from 1800 B.C.E. is the world’s first trigonometric table: using ratio-based trigonometry!

[Plimpton 322 is Babylonian exact sexagesimal trigonometry, Mansfield D., Wildberger N.J., 2017 Historia Mathematica]

Root systems (generalized)

A generalized (simply –laced) root system is a type of vector in an inner product space, each with the same quadrance, invariant under reflections in any associated perpendicular hyperplane, with reflections given by integral multiples of root vectors.

Theorem: R(X) for any graph X is a generalized simply- laced root system. It is finite precisely when X is ADE.

Root hyperplanes, Weyl chambers and orbits

A Weyl group W orbit for A2

Size of root populations and Weyl groups W = < s_x >

|R(E6)| = 36 + 36 = 72

|R(E7)| = 63 + 63 = 126

|R(E8)| = 120 + 120 = 240

|R(Dn)| = 2n - 2n

|R(An)| = n + n2

2

|W(An)| = (n + 1) !

|W(Dn)| = 2 . n!

|W(E6)| = 51,840

|W(E7)| = 2,903,040

|W(E8)| = 696,729,600

n

ADE ~ GraphsTheorem: The Tits quadratic form is degenerate precisely when X is an ADE~ graph.

Then spectral radius (X)= 2, and a Perron Frobenius vector has quadrance Q(v) = 0

Remove the ~ node, and you get the maximum root population on the associated ADE graph

Remarkable latticesIn the case of an ADE graph, P(X) is a (Euclidean) geometric lattice since the Tits quadratic form is positive definite. But these lattices also have remarkable properties!

Dim 1 2 3 4 5 6 7 8 > 8

Best Lattice Packing A1 A2 A3 D4 D5 E6 E7 E8 ??

LargestKissing Number

A1 A2 A3 D4 D5 E6 E7 E8 …

Number of sphereneighbours

2 6 12 24 40 72 126 240 …

An E7 Mutation Sequence

And an associated mutation frame… the swallow!

Another E7Mutation Sequence

An E8 Mutation Sequence

The associated E8 Mutation Frame

This is a pomset: a partially ordered multiset!

The swallow: the E7 X-heap

In “A Combinatorial Construction of simply-laced Lie algebras” (2003), I show how to construct ADE Lie algebras except E8 through minuscule representations via spaces of ideals on such heaps.

For E7 this gives the smallest 56 dim representation.

In another paper I give a similar realization of the 14 dim rep of G2.

Lie algebra representations and weights

15 dim representation of sl(3) or su(3)

Mutations, Root systems and related heaps/lattices/pomsets on general graphs X:

A huge area of potential investigation!

Thanks for listening, and many thanks to the organizers of G2 R2!

Join the Algebraic Calculus One course:n.wildberger@unsw.edu.au